Can I train a neural network incrementally given new daily data?
Yes, this is possible. Continuously extending your training data is known as incremental learning. 

You might also want to take a look at transfer learning, in which you reuse a trained model for a different purpose. This is very useful if you have a smaller dataset. 

In your particular case, you could train a NN once using your data from 2010 to 2019 and use it as a base model. Every time you get new data, you can use transfer learning to slightly re-train this model. Based on parameters such as the number of epochs and the learning rate, you can determine how much of an impact this new data will have.